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Related Concept Videos

Introduction to Test of Independence01:21

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In statistics, the term independence means that one can directly obtain the probability of any event involving both variables by multiplying their individual probabilities. Tests of independence are chi-square tests involving the use of a contingency table of observed (data) values.
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The test of independence is a chi-square-based test used to determine whether two variables or factors are independent or dependent. This hypothesis test is used to examine the independence of the variables. One can construct two qualitative survey questions or experiments based on the variables in a contingency table. The goal is to see if the two variables are unrelated (independent) or related (dependent). The null and alternative hypotheses for this test are:
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The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to conclude whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence. The hypotheses for the test for homogeneity can...
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Probability is the likelihood of an event occurring. The term event is defined as a collection of results of a procedure. An event is a simple event when an outcome cannot be divided into simpler parts.
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One-two dependence and probability inequalities between one- and two-sided union-intersection tests.

Helmut Finner1, Markus Roters2

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Summary
This summary is machine-generated.

This study generalizes the one-two inequality for union-intersection tests, providing bounds for acceptance/rejection probabilities. The findings offer insights into directional error control in multiple testing scenarios.

Keywords:
directional errorsgoodness of fit testsmultiple hypotheses testingone-two inequalitypositive association

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Area of Science:

  • Statistics
  • Probability Theory

Background:

  • The one-two inequality relates one- and two-sided coverage probabilities for empirical distribution functions.
  • Wald and Wolfowitz first discussed this inequality in 1939, later proved by Vandewiele and Noé for Kolmogorov-Smirnov tests.

Purpose of the Study:

  • Generalize the one-two inequality for union-intersection tests.
  • Introduce and analyze the concept of one-two dependence.

Main Methods:

  • Reviewing notions of positive association and related results.
  • Introducing and discussing the concept of one-two dependence.
  • Applying Bonferroni and one-two inequalities to derive bounds.

Main Results:

  • The one-two inequality is generalized for union-intersection tests with positively associated random variables.
  • Positive association implies one-two dependence, but not vice versa.
  • Bonferroni and one-two inequalities provide close bounds for acceptance/rejection probabilities.

Conclusions:

  • The generalized one-two inequality offers valuable bounds for statistical tests.
  • The concept of one-two dependence is introduced and contrasted with positive association.
  • The findings have implications for directional error control in multiple testing.